C₂×#2076723 #72299124

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#2076723

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

44207 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 22 points:

(12,14)(16,19)(20,22);(3,4,5,6)(11,12)(13,14)(15,16)(17,20)(18,19)(21,22);(8,10)(16,19)(17,21);(4,6)(7,8)(9,10)(15,16)(17,22)(18,19)(20,21);(3,4)(5,6)(15,17,18,21)(16,20,19,22);(1,2)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72282328, C₂×#72298074, C₂×#72298085
maximal normal subgroup of	C₂×#72282328, C₂×#72298074, C₂×#72298085

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((12,14)(16,19)(20,22), (3,4,5,6)(11,12)(13,14)(15,16)(17,20)(18,19)(21,22), (8,10)(16,19)(17,21), (4,6)(7,8)(9,10)(15,16)(17,22)(18,19)(20,21), (3,4)(5,6)(15,17,18,21)(16,20,19,22), (1,2));

C2×#2076723 #72299124